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To find the area between the curves \( y = x^2 \) and \( y = 2x \), we need to determine the points where the curves intersect, then compute the integral of the difference between the two functions over the interval between those points.

### Steps:
1. **Find the points of intersection**:
   Set \( x^2 = 2x \) to solve for \( x \).
   \[
   x^2 - 2x = 0
   \]
   Factor the equation:
   \[
   x(x - 2) = 0
   \]
   So, the points of intersection are \( x = 0 \) and \( x = 2 \).

2. **Set up the integral**:
   The area between the curves is given by:
   \[
   A = \int_{0}^{2} (2x - x^2) \, dx
   \]
   Here, \( 2x \) is the upper curve and \( x^2 \) is the lower curve.

3. **Compute the integral**:
   \[
   \int (2x - x^2) \, dx = \left( x^2 - \frac{x^3}{3} \right)
   \]
   Now, evaluate the integral from 0 to 2:
   \[
   A = \left[ x^2 - \frac{x^3}{3} \right]_{0}^{2}
   \]
   Substituting \( x = 2 \):
   \[
   A = \left( 2^2 - \frac{2^3}{3} \right) - \left( 0^2 - \frac{0^3}{3} \right)
   \]
   \[
   A = \left( 4 - \frac{8}{3} \right)
   \]
   Simplify:
   \[
   A = 4 - \frac{8}{3} = \frac{12}{3} - \frac{8}{3} = \frac{4}{3}
   \]

Thus, the area between the curves is \( \frac{4}{3} \) square units.

Would you like any further details or explanations? Here are 5 related questions to deepen your understanding:

1. How do you determine which function is the upper or lower curve?
2. What happens if the curves cross multiple times?
3. How do you compute the area if the functions are not simple polynomials?
4. What does it mean geometrically if the area is negative?
5. How would the process change if we were asked to compute the volume between two surfaces instead of area?

**Tip**: When solving for areas between curves, always check the intersection points carefully to ensure you're integrating over the correct interval.

Learn how to find the area between the curves y = x^2 and y = 2x with a detailed step-by-step integration process. This problem involves finding intersection points, setting up a definite integral, and calculating the area between the two curves.

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